Theta Function as a Section

For the line bundle from the previous slide, a typical holomorphic section is given by a theta function:

\[\vartheta(z,\tau)=\sum_{n\in\mathbb{Z}}\exp(\pi i n^2\tau+2\pi i n z).\]

This is not an ordinary function on the quotient torus.

Instead, it transforms in a controlled way under lattice translations, matching the transition rule of the line bundle.

In the picture, we ignore the size of $\vartheta(z,\tau)$ and color the complex plane by its phase:

\[\arg \vartheta(z,\tau).\]